A coupe of comments:
1. The countries with nuclear weapons have the capabilities and rights under the treaty to produce weapons-grade enriched uranium. And they all need to continue to do so to maintain their nuclear warheads, which have continuous natural nuclear decay going on that can degrade them. So, the issue is commercial use of weapons-grade material, not having it, and treaties against "weapons in space", which a rocket motor really is not.
2. Regarding using light weight atoms (or molecules) for the highest efficiency in producing rocket thrust, the reason is that rocket thrust is based on the momentum of the exhaust gas, while the speed of the exhaust gas depends on the amount of energy that can be put into it.
With a nuclear rocket, the energy is not dependent on the chemistry of the propellants, so the choice of exhaust gases is free of the need for it to be heated by burning it.
So, for example, if you apply enough energy to push an atom with atomic mass 1 to a velocity of 1, that amount of energy is 1/2 x mass x velocity squared which is 0.5 for those values.
Now consider a single exhaust gas molecule with a mass of 2. Assuming the same amount of energy in that exhaust particle and solving for its velocity, that particle will have a velocity equal to the square root of (0.5 divided by1/2 and divide by 2) = square root of 1/2 [instead of the square root of 1] = 0.71
Now, look at the momentum for the same amount of energy in the two particles of different masses. Momentum is proportional to the velocity, not the square of the velocity.
So, the particle with mass 1 and velocity 1 has momentum of 1, while the particle with mass 2 has momentum equal to 2 times 0.71 = 1.42. So, it seems like the heavier particle provides more thrust for a given amount of energy per particle. But, not for a given mass of propellant, which is the concern for rocket vehicle performance.
The efficiency ("specific impulse" abbreviated Isp) of a rocket motor is the amount of thrust per unit mass of the propellant used per unit time. So for the particle with mass 1 being used one per second, the Isp is 1.0 divided by one per second, which equals 1 second. The particle with mass 2 being used 1 per second has an Isp of 1.42 divided by 2 equals 0.71. so, the efficiency of the heavier particle is lower by 29%.
To really appreciate the significance of the Isp efficiency, think of it as the total amount of "impulse" that it can produce, which is the amount of force multiplied by the length of time it can be produced. So, for the 2 propellant particles of different masses, the lighter particle with an Isp of 1 second would produce a total "impulse" of 1 times 10 =10 for 10 seconds of propellant flow, while the heavier particle with the same mass flow rate would only produce 0.71 times 10 = 7.1 , because the same mass flow rate of the heavier propellant is 1/2 the particle flow rate of the lighter propellant.
So, for a given total mass of propellant, the lighter exhaust particles provide more total thrust over the same period of time that the rocket engine runs.
That is the basic physics, but the engineering is not so simple. For one thing, the weight of the equipment needed to create the propellant exhaust velocity is much different between a chemical rocket and a nuclear thermal rocket, which needs a whole, heavy nuclear reactor and probably some shielding that a chemical rocket does not have. And, there are also some other issues about where the energy from the reactor can go besides the velocity of the propellant particles. For instance, if the propellant particle is a molecule instead of a single atom, when it is heated, some of the energy goes into either vibrational and rotational states of the atoms within the molecule, or into separation of those atoms by breaking the chemical bond between them.
So, in reality, the benefits of a nuclear rocket vs a chemical rocket need to be calculated between 2 actual designs, one for each, to determine the real difference for specific mission requirements.